Center of masses and proving gravity

I am just curious to know what the advantages are to using center of mass for velocity are in a system of particles?

Also, how did newton prove that the motion of the planets around the sun was the same as a projectile on earth (I understand that this helped prove the force of gravity was responsible for both, but i dont understand how he proved them inherently the same)

The sum of the momenta of all the particles is zero in the CM frame, which simplifies calculations in many cases. The total angular momentum or KE of a system of particles is equal to the angular momentum or KE of the total mass sitting at the CM plus the angular momentum or KE about the CM. All sorts of stuff like these make the CM frame very useful.

Newton (and a few others) hypothesized that it was the same force that moved the planets around the Sun that pulled apples down from trees. He didn’t prove it. He extended this property to all matter in the Universe. We can't quite make out the enormity and grandeur of the statement sitting here in the 21st century. Then he developed the mathematics for it, which other people couldn’t. A lot of things sort of made more sense after that using his law.

jr writes>
Newton defined centripetal force in terms of his second and third law by setting his first law object on a uniform circular path of motion, where the law of areas falls out as an efficient area enclosing artifact of the circle itself. He then connected the efficient time artifact of the circular motion to Kepler's efficient time controlled law of areas and generalized Kepler's law of areas to the entire universe as the mathematical carrier for his centripetal force. He assigned the resistance he worked against (inertial mass) and called gravity as the "a priori" cause of the centripetal force.

If we define gravity as the resistance we work against and quantify in terms of our inertial mass, or the inertial mass of other inertial mass objects that we as inertial mass objects interact with. an equal and opposite quantitative, but nonetheless subjective notion of force ensues. This is proved gravity. However it is not proved to act between celestial objects or between inertial mass objects and celestial objects.
Have a good time.
johnreed